 Our sixth group is on the topic of billiards with a twist. So the advisor is Jaidev Athraya, and the three presenters are Nicholas Cecil, Thibaut Langley, and Michael Ouijaia. Okay, so we're presenting billiards with a twist. Wrong direction. So we first started with billiards in the square. So this is where you have a physically ideal billiard table where there's no friction, there's no gravity, and collisions are perfectly elastic. And you start with a billiard ball with a known position and velocity and wonder what happens over time. It turns out that if the slope of the line is rational, the orbit of that billiard ball is periodic. It'll repeat eventually. And if the slope is irrational, it won't repeat. And in fact, in the irrational case, you find that the collision points are dense and equidistributed around the square. So we moved to considering what happens in a circle. But the twist is that instead of bouncing in the normal physical sense, you hit the billiard ball with the transform one over z. And we wonder what happens, what happens over time? Do you get the same results on periodicity or do you get density, that kind of thing? So to explain our notation, when the point is on the disc or when it's on the unit circle, you can just describe it with theta and alpha where theta is the angle from the horizontal of the position and alpha is the angle from the horizontal of the velocity. So that's what that diagram here shows. And the idea is the impact point in the first quadrant is the first thing that happens and then the one over z transform moves you down to the impact point in the second, sorry, in the fourth quadrant and then it flows across the circle. And these are the results, what happens to your initial point in terms of theta and alpha. So we started with a simple case, the radial case, where the billiard ball starts in the center and moves radially. And as you can see, the billiard ball eventually ends up just where it started with the same velocity. And that means it's periodic and it's periodic for any angle. So then we moved to a more general case where you don't know at the beginning where you start and what your velocity is. It could be anywhere. And so it turns out that this creates a relatively simple geometric problem that you can solve where the billiard ball end up after it transforms and flows across the circle along its velocity. And some simple calculations show that you get this rule and that turns out to be an affine map. Which is kind of nice because we start with this weird system. You don't know what's going to happen. And it turns out to be this relatively nice equation. So, okay. So we can get just the solution to the evolution by brute force matrix calculation. But actually I hate calculation so I will show you how you can get this without any calculation. And just some geometry. So first we just rewrite our system in this form. And what you can see is that there are two parts in my evolution. First part I just make a symmetry with respect to y-axis. As second part I get just a rotation of angle to alpha n minus theta n. So and it's the same evolution for speed and position. So why is it useful? So let us imagine that, okay, I start with this. And this system says that first I make a symmetry. Okay, sorry, symmetry with respect to y-axis. And then I just make a rotation with the angle. But not to understand what happens, I will do something weird. Is that I will just go to the other side of the board and oh imagine it's transparent and I can just see. What I see is that the next position is here. And what you get is just a rotation of angle to alpha minus theta. And also note that in real life the difference of angle between speed and position is just minus like the opposite of what we had before. But no, because I've gone to the other side of the board is just exactly the same than before. So yeah, because maybe you're not really convinced with this, get some pictures. And so I just start with some random initial position and speed. And here is the evolution if you're always in the same side of your board. But let us know like, okay, on the left you just have just the same thing like, okay, evolution and you just stay in the same side of the board. And here is what I see on the right when I just, each time I just change side of the board and look at what happens. And oh, you just see a rotation on the right side. So, yeah, this is basically why you are like, the rotation part is this part like, it's what you get if you have a rotation, okay. And the word term with the minus one to the power n is just like, okay, when we, when n is an odd number I just also have a symmetry and I wanna have also pi here. So I try to have a formula for saying that, okay, for even numbers I just get a rotation and for odd number I also have a symmetry. And the same thing for the speed which explains like, okay, it looks pretty complicated but it's just a rotation composed with a symmetry. So, okay, now we can try to figure out what are the dynamical properties of our system. And actually, because it's like just a rotation within some, sometimes a symmetry, we just have the same, basically, the same characterization of periodic orbits. Periodic orbits is just when the angle of the rotation over pi is a rational number. So actually we can even figure out and compute the period but I think I don't have time to explain because like, got other things to do but actually it's not really easy but you can compute and figure out what is the period. So now we just turn to the case in which the angle of rotation over pi is not rational. And so the result, if we adjust a rotation would be my orbit is dense on the circle and even equidistributes. And actually the same results stands here basically because you can just look at the two sequence theta2n and theta2n plus one and see that you just got rotation and so the density and equidistribution still stands. Okay but now we can try to look at the actual trajectory because for the moment we just have looked at the deposition in which the ball hits the boundary but we can try to draw the actual trajectory and ask for example whether the trajectory would be dense in the interior of the unit disk. Okay that is my question, that is your answer, it's no. And it's quite easy to see that just because of this is that I don't know if you remember how we compute the next position but we had something like this and the angle beta here was just theta minus alpha and because at minus one it's just the same in the evolution you always have the same angle here which means that basically your trajectory is contained between two circles. So now here is the new answer, is that okay but is the actual trajectory dense between these two circles and here is the answer, this time is yes and how to see that is like okay, sorry. Yeah yeah okay I'll just conclude like. Okay here I take one random point between these two circles and draw an epsilon neighborhood and I'm like oh what are the trajectory for which I can just go into the epsilon neighborhood and so here in the right part of the circle is one like if I bounce in the right part I'm sure because of the drawing that I will go into the epsilon neighborhood and because my orbit is dense in the circle I'm sure that I will go there. So basically I got density. So yes we want to thank Jayadev for his advice and I will just conclude by saying that sometimes it can be helpful to see a program from the other side of the world. Thanks.